Due to the very different energy and time scales involved in prompt charmonium production, phenomenological models assume that the cross section factorises into a hard term, describing the initial production of the \(\mathrm {c\overline{c}}\) pair, and a soft term accounting for the subsequent evolution into a bound state. While the production of \(\mathrm {c\overline{c}}\) pairs can be computed within perturbative QCD, their evolution to a bound state involves long-distance physics which are non-perturbative. Their determination relies largely on fits to experimental measurements. A detailed overview of this field of study can be found in Refs. [1]}, [2]}, [3]}. There are a few different approaches employed for the description of quarkonium production, namely the Colour Singlet Model (CSM) [4]}, the Colour Evaporation Model (CEM) [5]}, [6]}, and the Non-Relativistic QCD model (NRQCD) [7]}.
While the CSM model is known to underestimate the production cross sections [8]}, both the NRQCD and improved CEM models provide a better description of the measured cross sections [9]}, [10]}, [11]}. However, the simultaneous description of the differential production cross sections and the charmonium state polarisation is still not achieved [12]}, [13]}, although recent calculations within the \(k_{\mathrm {T}}\) -factorisation approach seem to improve the agreement with polarisation measurements [14]}.
